Working Report 2002-39

Integration of equivalent continuum and discrete fracture network approaches

in groundwater flow modelling - Preliminary study

Jari Lofman

VTT Energy {now VTT Processes)

Ferenc Meszaros

The Relief Laboratory, Harskut, Hungary

August 2002

Working Reports contain information on work in progress

or pending completion .

The conclusions and v iewpoints presented in the report

are those of author(s) and do not necessarily

coincide with those of Posiva.

Working Report 2002-39

Integration of equivalent continuum and discrete fracture network approaches

in groundwater flow modelling - Preliminary study

Jari Lofman

Ferenc Meszaros

August 2002

-tTr ENERGY

Research organisation and address Customer

VTT Energy, Nuclear Energy Posiva Oy P.O. Box 1604 Toolonkatu 4 FIN-02044 VTT, FINLAND FIN-00100, Helsinki, FINLAND

Contact person Contact persons

Jari LOfman Aimo Hautojarvi

Diary code Order reference

ENE4-51 T-2000 9703/00/ AJH (26.1 0.2000)

Project title and reference code Report identification & Pages Date Integration of Equivalent Continuum and Discrete Fracture ENE4/28/01 19.12.2001 Network Approaches in Groundwater Flow Modelling -

Jl:.+Appendices(6p.~~ u _ Preliminary Study (NOSU00403, 45INTEGOO) f/11: 7.f. ZIJtJ2.. . un1d ~A'-, /

Report title and author(s)

Integration of Equivalent Continuum and Discrete Fracture Network Approaches in Ground water

Flow Modelling - Preliminary Study

Summary

This work comprised a preliminary study on the integration of site and canister scale modeh The integrated model consists of both the equivalent continuum (EC) and discrete fracture n of flow and conservation of mass at the interface between the EC and DFN medium ensure t canister scale are consistent with the localised conditions on the (larger) site scale.

l.: Q K..l "- c_, Rt:. I ,. -eT

((.__-~ 7

The main objective of this work was to develop an integrated EC-DFN model (first as a pilot project in 2D with special emphasis placed on the extensibility of the code to 3D), in which the two models would be connected at the mesh level. The new approach was based on a previously developed FEFTRA/ quad tree code, which employs an adaptive and recursive algorithm in mesh generation. The algorithm's capability of efficiently incorporating local refinements in the mesh enabled a natural and straightforward coupling of the two different models at the mesh level at their common nodes, virtually eliminating the problems associated to data transfer across the EC-DFN interface. The developed integrated model was applied to one demonstration case based on the latest groundwater flow analyses for the Olkiluoto site. The results indicated that the integrated model and FEFIRA finite element code could handle the realistic cases that could be encountered in foreseeable modelling tasks. Connecting the two submodels on the mesh level proved to be a viable and efficient alternative to the current solutions.

The study also consisted of a site scale model testing in which a steady-state groundwater flow in 3D was computed with NAMMU and FEFfRA program packages for the Olkiluoto site. The comparison of the results showed an excellent agreement between the FEFfRA and NAMMU models, which verified further the capability of the FEFIRA code to simulate a real-life site-scale groundwater flow problems employing the 2D elements for the fracture zones in the 3D mesh.

AEA Technology performed preliminary flow simulations for the Olkiluoto site by applying an integrated site and canister scale models. The simulations were carried out by the finite element program CONNECTFLOW, which couples the EC and DFN approaches at the equation level. In this work, the principles of the CONNECTFLOW model were also reviewed.

Princi~ut~

1 ar~fman, Research Scientist, Nuclear Waste Management

A~r~~:d by ~~ ~ori, Research Manager, Nuclear Energy

Re'rfrd

HelLa~aiko, Group Management Availability statement

To be published in Posiva's report series

The use of the name of the Technical Research Centre of Finland (VTT) in advertising or publication in part of this report is only permissible by written authorisation from the Technical Research Centre of Finland

L ~

Integration of equivalent continuum and discrete fracture network approaches in

groundwater flow modelling - Preliminary study

Jari Lofman VTT Energy

Ferenc Meszaros The Relief Laboratory

19th December 2001

----------------------------~--

INTEGRATION OF EQUIVALENT CONTINUUM AND DISCRETE FRACTURE NETWORK APPROACHES IN GROUNDWATER FLOW MODELLING - PRE­LIMINARY STUDY

ABSTRACT

This work comprised a preliminary study on the integration of site and canister scale models into a single ground water flow model. The integrated model consists of both the equivalent continuum (EC) and discrete fracture network (DFN) medium. The continuity of flow and conservation of mass at the interface between the EC and DFN medium ensure that the flow conditions on the (smaller) canister scale are consistent with the localised conditions on the (larger) site scale.

The main objective of this work was to develop an integrated EC-DFN model (first as a pilot project in 2D with special emphasis placed on the extensibility of the code to 3D), in which the two models would be connected at the mesh level. The new approach was based on a previously developed FEFTRA/ quad tree code, which employs an adaptive and recursive algorithm in mesh generation. The algorithm's capability of efficiently in­corporating local refinements in the mesh enabled a natural and straightforward coupling of the two different models at the mesh level at their common nodes, virtually eliminating the problems associated to data transfer across the EC-DFN interface. The developed integrated model was applied to one demonstration case based on the latest groundwater flow analyses for the Olkiluoto site. The results indicated that the integrated model and FEFfRA finite element code could handle the realistic cases that could be encountered in foreseeable modelling tasks. Connecting the two submodels on the mesh level proved to be a viable and efficient alternative to the current solutions.

The study also consisted of a site scale model testing in which a steady-state ground­water flow in 3D was computed with NAMMU and FEFfRA program packages for the Olkiluoto site. The comparison of the results showed an excellent agreement between the FEFfRA and NAMMU models, which verified further the capability of the FEFfRA code to simulate a real-life site-scale groundwater flow problems employing the 2D ele­ments for the fracture zones in the 3D mesh.

AEA Technology performed preliminary flow simulations for the Olkiluoto site by apply­ing an integrated site and canister scale models. The simulations were carried out by the finite element program CONNECTFLOW, which couples the EC and DFN approaches at the equation level. In this work, the principles of the CONNECTFLOW model were also reviewed.

Keywords: ground water flow, integrated model, discrete fracture network, equivalent con­tinuum, bedrock, repository, nuclear waste, numerical modelling

ALUSTAVA SELVITVS HUOKOISEN VALIAINEEN JA RAKOVERKKOMALLIEN VHDISTAMISESTA POHJAVEDEN VIRTAUSANALVVSEISSA

TIIVISTELMA

Tyossa tarkasteltiin tutkimusalueenja loppusijoituskanisterien mittakaavan mallien yhdis­tamisUi. pohjaveden virtausanalyyseissa. Yhdistetty malli sisaltaa seka huokoista (equi­valent continuum, EC) valiainetta etta rakoverkkoa (discrete fracture network, DFN). Virtauksen jatkuvuus ja massan sailyminen EC- ja DFN-valiaineiden valisella rajapin­nalla takaavat, etta yhdistetyssa mallissa virtausolosuhteet tutkimusalueen ja loppusijoi­tuskanisterien mittakaavassa ovat konsistentit.

Tyon paaasiallisena tavoitteena oli kehittaa yhdistetty malli (ensin kokeilumielessa kaksi­ulotteisena kiinnittaen samalla erityista huomioita mallin laajentamiseen kolmiulotteisek­si), jossa EC- ja DFN-valiaineet on kytketty toisiinsa elementtiverkon tasolla. Uusi mene­telma perustui aikaisemmin kehitettyyn FEFTRA/ quad tree ohjelmaan, joka soveltaa adaptiivistaja rekursiivista puualgoritmia verkon generoinnissa. Elementtiverkon tehokas paikallinen tihentaminen mahdollisti kahden eri mallin suoraviivaisen yhdistamisen raja­pinnalla olevien yhteisten solmujen kautta. Menetelmalla voidaan valttaa vaihtoehtoisiin lahestymistapoihin liittyvat ongelmat rajapinnan kasittelyssa. Kehitettya menetelmaa so­vellettiin yhteen Olkiluodon viimeisimpiin virtausanalyyseihin perustuvaan esimerkkita­paukseen. Tulokset osoittivat, etta yhdistetylla mallilla ja FEFfRA-ohjelmistolla voidaan laskea realistisia ja kayHi.nnollisia eri mittakaavan mallit sisaltavia tapauksia. Kahden eri mallin yhdistaminen toisiinsa elementtiverkon tasolla osoittautui hyvaksi ja tehokkaaksi vaihtoehdoksi muihin vastaaviin menetelmiin.

Tama raportti sisaltaa my os tutkimusalueen mittakaavassa suoritetun testitapauksen, jossa Olkiluodon tutkimusalueelle laskettiin kolmiulotteista pohjaveden virtausta tasapainoti­lassa kayttaen NAMMU- ja FEFfRA-ohjelmistoja. Kahdella eri ohjelmistolla lasketut tu­lokset vastasivat erinomaisesti toisiaan, mika edelleen varmisti seka FEFfRA-ohjelmiston soveltuvuuden todellisiin tutkimusalueen mittakaavassa suoritettaviin pohjaveden virtaus­analyyseihin etta kaksiulotteisten elementtien kayton ruhjevyohykkeiden kuvaamiseen kolmiulotteisessa elementtiverkossa.

AEA Technology on suorittanut alustavan pohjaveden virtausanalyysin Olkiluodon tut­kimusalueelle kayttamalla yhdistettya tutkimusalueen- ja loppusijoituskanisterien mit­takaavan mallia. Simuloinneissa kaytettiin CONNECTFLOW-ohjelmistoa, joka yhdistaa yhtalotasolla huokoisen valiaineen ohjelmiston NAMMU ja NAPSAC-rakoverkkomallin­nusohjelmiston. Tassa tyossa myos selvitettiin CONNECTFLOW-ohjelmiston sovelta­maa menetelmaa eri mittakaavan mallien yhdistamisessa.

Avainsanat: pohjaveden virtaus, yhdistetty malli, rakoverkko, huokoinen valiaine, kallio­pera, ydinjate, numeerinen mallinnus

1

TABLE OF CONTENTS

Abstract

Tiivistelma

Preface................................................................... 3

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 SITE-SCALE TESTING AT THE OLKILUOTO SITE.. . . . . . . . . . . . . . . . . . . . . . 7 2.1 Description of the case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Numerical solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 INTEGRATED EC-DFN MODEL IN CONNECTFLOW..................... 17 3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 DFN model equations . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 EC model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4 Interface conditions- mass lumping approach. . . . . . . . . . . . . . . . . . . . . . 19 3.5 Interface conditions- distributed flux approach . . . . . . . . . . . . . . . . . . . . . 19

4 QUADTREE-BASED INTEGRATED EC-DFN MODEL IN FEFTRA . . . . . . . . . 21 4.1 Review of current solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Objectives and basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Necessary input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 The process of generating the DFN submodel . . . . . . . . . . . . . . . . . . . . . . . 25 4.5 Connecting the DFN submodel to the EC model . . . . . . . . . . . . . . . . . . . . . 29 4.6 Computational aspects, implementation details & QA . . . . . . . . . . . . . . . . 30

5 EXAMPLE OF THE QUADTREE-BASED INTEGRATED EC-DFN MODEL . . 31

6 SUMMARY AND DISCUSSION .. . . .. .. .. . . . .. .. . .. . .. .. .. .. . .. .. .. . . .. 39

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

APPENDICES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3

PREFACE

The study was carried out by VTT Energy on contract for Posiva Oy. The extension of the mesh generation program quadtree to the integrated equivalent continuum-discrete fracture network modelling (Chapter 4), which comprised the most demanding part of the work, was implemented by The Relief Laboratory, Hungary, on subcontract for VTT En­ergy. On behalf of Posiva the work has been supervised by Dr Aimo Hautojarvi, whereas Mr J ari Lofman was the contact person at VTT Energy and Mr Ferenc Meszaros at The Relief Laboratory.

5

1 INTRODUCTION

Ground water flow modelling has been an essential part of the site investigation work car­ried out for the final disposal of spent nuclear fuel in crystalline bedrock in Finland. For the latest safety analyses TILA-99 (Vieno & Nordman 1999) the site-specific ground­water flow analyses at Olkiluoto were carried out by applying two nested and separate models on different scales. The equivalent continuum (EC) approach was employed on the site scale (Lofman 1999) and the discrete fracture network (DFN) approach on the canister scale (Poteri & Laitinen 1999). The objective of the site scale modelling was to characterise the overall groundwater flow conditions and provide general insight about the mechanisms affecting the flow during the next few thousand years. The purpose of the canister scale modelling was a more detailed examination of the groundwater flow con­ditions and the transport properties in the vicinity of the disposal canister and along the potential release paths from the repository into the biosphere. The two separate models had a very simple one-way coupling, i.e. the overall flow properties for the canister scale were derived from the results of the site scale modelling. When transferring flow condi­tions from site to canister scale the average behaviour of larger scale flow was considered to offer reasonable boundary conditions to the canister scale model. However, as the size and modelling approaches between the two models were quite different, the estimation of the boundary conditions for the DFN model was subject to inconsistencies and large uncertainties.

The objective of this work is to carry out a preliminary study on the integration of site and canister scale models into a single groundwater flow model. The integrated model consists of both the EC and DFN media. The continuity of head/pressure and conservation of mass at the interface between the EC and DFN media ensure that the flow conditions on the canister scale are consistent with the localised conditions on the site scale.

Hartley et al. (200 1) performed flow simulations for the Olkiluoto site by applying mod­els on different scales. The study consisted of a testing of non-integrated models on a site and canister scale as well as a simulation employing an integrated model. In the site scale model testing, a steady-state, three-dimensional groundwater flow (with no consid­eration of salinity) was computed with the NAMMU (NAMMU 2001) and the FEFTRA (FEFTRA 2001) program packages for the Olkiluoto site and the results were compared. In this report, the modelling efforts with the FEFTRA package are documented. In the integrated simulations Hartley et al. (2001) used the CONNECTFLOW program (Hartley 1996; Holton & Milicky 1997), which combines a site scale EC and canister scale DFN models at the equation level. In this report, the CONNECTFLOW approach is reviewed on the basis of the available material.

The main objective of this work was to develop an integrated EC-DFN model (first as a pilot project in 2D with special emphasis placed on the extensibility of the code to 3D), in which the two models would be connected at the mesh level. During 1998-2001 an advanced mesh generation program was developed for the FEFTRA code. The program is based on the adaptive and recursive quadtree/octree algorithm, which can easily be em­ployed to refine mesh near desired locations (e.g. fracture zones, tunnels, shafts, sinks,

6

etc.) or at high gradients of any computed result quantity, even in transient problems. At the moment the 2D version of the program (FEFTRA/ quad tree) has been imple­mented, tested and finished, while the 3D version ( octree) is still under development. As the algorithm enables efficient local refinement of the mesh without high computational costs, it was considered to be most applicable also to the integration of the DFN subregion into the EC model. Especially, the tree method enables a natural and straightforward cou­pling of the two different models at the mesh level with their common nodes at the inter­face between the two submodels. Within this work, the 2D version of the integrated model is developed for FEFfRA. The model is based on the FEFTRA/ quad tree code and it consists of the stochastic lD fracture network with stochastic distribution of the hydraulic properties integrated to the sunounding 2D equivalent continuum mesh. The model and its development is documented in detail, and it is applied to one realistic demonstration case based on the latest site and canister scale analysis for the Olkiluoto site.

This report is organised in the following manner. The site-scale testing between the FEFfRA and NAMMU models for the Olkiluoto site is presented in Chapter 2. The integrated CONNECTFLOW model is introduced in Chapter 3, while the documentation of the corresponding quad tree-based FEFfRA model is presented in Chapter 4. An example of the developed quad tree-based integrated model is introduced in Chapter 5. Summary and discussion are given in Chapter 6.

7

2 SITE-SCALE TESTING AT THE OLKILUOTO SITE

Hartley et al. (200 1) performed flow simulations for the Olkiluoto site by applying both non-integrated and integrated models on a site and canister-scales. Especially, the study consisted of a site-scale model testing, in which a steady-state, three-dimensional groundwater flow with no salinity assumed was computed with the NAMMU (NAMMU 2001) program package for the Olkiluoto site. In this chapter, the corresponding FEFfRA (FEFfRA 2001) simulations are presented and the results are compared against those obtained with NAMMU.

This is the first time that the FEFfRA package in its present form is compared to a similar code with a real-life, site-scale groundwater flow problem. Particularly, as the fracture zones in the 3D models are represented by 3D elements in NAMMU and 2D elements in FEFfRA, the performance of the differing representations of the zones in the two packages can be assessed with this case.

2.1 Description of the case

The case was comprised of a steady-state, site-scale simulation of groundwater flow assuming freshwater conditions. As the site-specific flow model is practically identical to the one employed in the latest study by Lofman (1999), most of the details are omitted here and only a summary and the modifications to the previous model are presented.

The size of the modelled bedrock volume was 6.3 km x 4.3 km x 1.5 km (Figure 2.1 ). The modelled area and the inner refined area (the area covered by the borehole investiga­tions) were slightly enlarged compared to (Lofman 1999). The modelled bedrock volume was conceptually divided into intact (i.e. sparsely fractured) rock and fracture zones (Fig­ure 2.2). The geometry of the 33 planar zones was based on a revised bedrock model by Saksa et al. (1998). The following slight modifications were made to some of the zones compared to (Lofman 1999):

• AR3, AR6, AR8, Rl, R4, R4I, R5 and R27 were extended to the outer boundary of the model

• AR5 was excluded, because of the location near R5

• ARl, AR6, Rl, R4, R7 were slightly simplified

• R6L was included in the name R6, and R18L in R18 (no changes)

• zone R28A and R28B renamed to R28AB (no changes)

The equivalent-continuum model was applied separately for the intact rock and each frac­ture zone, which were assumed to consist of five hydrological layers (Tables 2.1- 2.3). In recent reports (Aikas et al. 2000) the approach has appeared to be to divide the rock into a number of depth classes, and calculate the properties for each depth class independently.

8

Hartley et al. (2001) considered that approach to be more compatible with the five-layer model than the functional form used in the latest study by Lofman (1999). The motivation for the particular choice of five layers was to approximate as closely as possible both func­tional form and the depth classes. The conductivity contrast between the two approaches in the different layers is less than an order of magnitude. The properties in the five layer model were calculated using a geometric mean for the conductivity/transmissivity and an arithmetic mean for the porosity.

The internal structure of the repository located at a depth of 500 metres was not considered in detail, but the repository was modelled as a two-dimensional structure (Figure 2.3) based on the layout by Lofman (1999). Specified pressure corresponding the water table (where water table data was available) or a simple linear transformation of the topography (watertable = 0.56 ·topography, where water table data was not available) was applied on the surface (Figure 2.4), and no flow on other boundaries.

SCALE

LEGEND

- 11 CLASS LINEAMENT

- Ill CLASS LINEAMENT

IV CLASS LINEAMENT

Base map:

'NationaiLand&lrveyoiFiriand, pennlulon 13/MYY/91

Figure 2.1. Outlines of the modelled area (large rectangular with dashed lines) and the inner refined area (small rectangular with dashed lines) on the ground surface.

------------------------- - ··· ·-

Up

LEast

9

Boundary of modelled volume

= -1500m

Figure 2.2. Conceptual fracture zone geometry for the Olkiluoto site.

Table 2.1. Classification of fracture zones by transmissivity.

Class Fracture zones A R3, R4, R5, R6, R7, R19HY, R20HY, R21, R24HY, R25, R27, R28, R29,

AR-zones B R1, R2, R8, R9HY, R10HY, R11, R16, R17HY, R18, R22, R23

2.2 Mathematical model

The flow equation in a steady state is written for the residual pressure Pr [Pa] (the to­tal pressure without the hydrostatic component of fresh water) as follows (Bear 1979; Huyakom & Pinder 1983; de Marsily 1986)

\1· (p: (VPr + (p- Po)g\lz)) = 0, (2.1)

where p is the density of water [kg/m3], p0 is the density of the fresh water [= 998.6

kg/m3], k is the permeability tensor of rock [m2

], f-L is the dynamic viscosity of water[= 1.0·10- 3kg/m/s], g is the gravitational acceleration[= 9.81 m/s2

], and z is the elevation relative to the sea level [m].

The velocity required in the flow path calculations is expressed in terms of the residual pressure as follows

(2.2)

10

Figure 2.3. The layout for the repository located at a depth of 500 meters.

Table 2.2. Transmissivity of fracture zones [m2 /s] and hydraulic conductivity of intact rock [m/s] . The transmissivity of 5.0 · 10-8 [m2 /s] was used for the repository located at a depth of 500 metres. The thickness of the zones and the repository are 10 and 5 m, respectively.

Depth [m] Zones (class A) Zones (class B) Intact rock 0 -lOO 2.53. 10-4 5.75. 10-6 2.88. 10-9

100- 200 9.14. 10-5 2.o8. 10-6 1.04. 1o-9

200-400 2.53. 10-5 5.74. 10-7 2.87. 10-10

400-900 3.18. 1o-6 7.23. 10-8 3.61. 1o-n 900- 1500 5.87. 10-7 1.33. 10-8 6.66. 10-12

where cjJ is the flow porosity [-],

The permeability k in Equations (2.1) and (2.2) is related to the hydraulic conductivity K [mls]

2.3 Numerical solution method

k = !!_K. pg

(2.3)

The finite element method with linear elements was applied in solving the case numeri­cally with FEFfRA. All 33 fracture zones were included explicitly in the modelled vol­ume, which was meshed with 536536 hexahedral elements for intact rock and 105529 triangular/quadrilateral elements for the fracture zones and the repository (Figure 2.5).

11

Table 2.3. Flow porosity [-] of the fracture zones and hydraulic conductivity of the intact rock. The porosity for the repository is 8.54 ·10-4

. Although also the values for the zones and repository are presented in the table, only the values for the intact rock were used to compute the travel times for the flow paths (see section "Numerical solution method").

Depth [m] 0-100 100- 200 200-400 400- 900 900- 1500

Zones (class A) 1.75. 10-2

1.14. 10-2

6.59. 10-3

2.51. 10-3

8.38. 10-4

Zones (class B) 5.02. 10-3

3.26. 10-3

1.88. 10-3

7.12. 10-4

2.37. 10-4

Intact rock 5.19. 10-4

3.36. 10-4

1.93. 10-4

7.33. 10-5

2.44. 10-5

elevation above sea level [m]

11

10

9

8

7

6

5

4

3

2

1

0.01

0

Figure 2.4. The present groundwater table of the Olkiluoto area on the surface of the model.

12

The mesh was created by embedding the 2D elements on the faces and/or diagonals of the 3D elements in the existing base mesh consisting of 3D elements only. Due to the (finite) element size of the base mesh, the surfaces representing the fracture zones in the mesh (Figure 2.7) are somewhat stepped compared to the corresponding planes defined in the bedrock model (Figure 2.2). As the 2D elements did not have physical thickness, the thickness of the zones and repository was assigned in the transmissivity of Equations (2.1) and (2.2). The size of the 3D elements in the inner refined area at the centre of the island was 27 m x 27 m x 25 m down to a depth of 1000 metres.

The partial differential equation (2.1) describing groundwater flow was solved numeri­cally employing the conventional Galerkin technique (Huyakom & Pinder 1983). The linear matrix equations resulting from the finite element formulation of Equation (2.1) was solved employing the conjugate-gradient method (Atkinson 1988; Laitinen 1994). The flow paths were computed with the algorithm that uses the continuous velocity field solved by applying the finite element method to Equation (2.2).

In the NAMMU finite element model the fracture zones are represented implicitly (Hart­ley et al. 2001) by manipulating the hydraulic conductivity tensor of the 3D elements crossed by the planes of the fracture zones. The apparent inconsistency arising from the representation of the zones (of thickness 10 m) by 3D elements (of size about 25 m) was resolved by assigning such an average permeability to the elements crossed by the zones, which ensures that the flow through the elements is correct. However, the porosity was not averaged, but the porosity of the intact rock was assigned to all the elements (whether crossed by the zones and repository or not).

The finite elements around the repository in the NAMMU mesh were of size about 25 m by 27 m by 5 m (Hartley et al. 2001 ). The horizontal discretisation is similar to the FEFfRA mesh, but there is greater vertical discretisation in the NAMMU model. In particular: the vertical element size is 10 m from -400 m to -480 m and 5 m -480 m to -520 m, while below -520 m a graded discretisation was used, which starts with finite elements of size 5 m increasing to elements of size about 20 m. The NAMMU mesh consisted of 863460 elements. Similarly to FEFfRA, a preconditioned conjugate gradient method method was used to obtain the solution for groundwater flow.

2.4 Results

The result quantities computed in the test case were

• pressure along the boreholes KR1-KR5,

• flow paths starting at three points near the repository,

• flow rates through a box surrounding the repository (the box is intersected by the zones R10HY and R16), and

• infiltration based on the computed flow rate through a horizontal plane at a depth of 10 metres.

t 1500m

!

Up

LEast

13

Figure 2.5. Finite element mesh for the Olkiluoto site in the FEFTRA model. The three­dimensional elements (536536) represent the intact rock (top) and the two-dimensional elements (105529) the fracture zones (bottom). Compare to Figure 2.2.

14

Table 2.4. Computed net flow rates [m3/a] through a box surrounding the repository ( + denotes inflow and - outflow). The height of the box is 50 metres and the area of the top and bottom faces is 800000 m2

, whereas the distance between the repository and the faces of the box is about 20-30 metres. The box is intersected by the zones Rl OHY and R16. The inaccuracy of mass balance in the NAMMU flows results from the fact that the box was not coincided with .finite elements (Hartley et al. 2001).

Face FEFfRA NAMMU Bottom face -24.98 -21.82 South-west side 0.58 0.58 South-east side 0.53 0.70 North-east side -0.11 -0.20 North-west side -0.60 -1.48 Top face 24.51 21.64 Mass balance -0.07 -0.58

The pressure along the boreholes are presented in Figure 2.6 showing an excellent agree­ment between the FEFfRA and NAMMU results. The computed net flow rates through a box surrounding the repository (Table 2.4) are also essentially in good agreement. The differences on the north-west and north-east side of the box can likely be attributed to the different representations of the fractures zones in the two models, which results in the northern corner of the box to clip several fracture zones in the NAMMU model and gives different local flows in this area (Hartley et al. 2001). The computed infiltrations (34.3 mm/a in the FEFfRA and 33.5 mm/a in the NAMMU model) are in excellent agreement.

The final positions and lengths of the flow paths (Table 2.5 and Figure 2.7) show a pretty good agreement between the FEFfRA and NAMMU models. However, the travel times of the flow paths computed with FEFfRA are shorter than those obtained with NAMMU. After leaving the repository the NAMMU paths (especially path 1 and 3) stay much longer in the intact rock than the FEFfRA paths. Due to the low velocity of the intact rock, a small change in the path through the intact rock may lead to a disproportionately large change in the travel time. Thus, the times are sensitive to the behaviour of the flow field in the intact rock between the repository and the nearest zones. The discrepancies in the local flow field, on the other hand, can probably be attributed to the different representations of the fracture zones and/or different discretisation in the vicinity of the repository (see section "Numerical solution method" above). Especially, the implicit representation of the zones in the NAMMU model may overestimate the travel times, because the permeability of the zones was reduced by averaging the permeability over the volumes of the finite elements crossed by the zones (Hartley et al. 2001).

The results proved that the representation of the fracture zones by lower dimensional elements (2D elements in the 3D mesh) is a feasible and efficient alternative to the use of uniform dimensional elements.

15

KRl KR2 0 0

• -200 -200 • I

,......., -400 ,......., -400 I g g N -600 N -600 •

• -800 -800

-1000 -1000 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70

p [kPa] p [kPa]

KR3 KR4 0 0

-100 • -200

,......., -200 • ,......., -400

s s N -300 N -600 •

-400 -800

-500 -1000 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70

p [kPa] p [kPa]

KR5 0

-100 - Measured values

,......., -200 g NAMMU

N -300

FEFTRA -400

-500 0 10 20 30 40 50 60 70

p [kPa]

Figure 2.6. Measured and simulated residual pressure along the cored boreholes KRJ­KR5 at the Olkiluoto site. The effect of salinity is included in the measured values, but not in the simulated values.

16

Table 2.5. Flow paths computed with FEFTRA and NAMMU.

Initial position [m] x y z

Final position [m] x

Pathlength [m] Travel time [a]

up

no~ west

y z

Path 1 5596 2288 -500 5683 3526

0 1677.7 1321.0

FEFfRA Path 2 5596 2288 -475 5680 3545

0 1680.4 1097.8

NAMMU Path 3 Path 1 Path 2 5596 5596 5596 2288 2288 2288 -525 -500 -475 5677 5697 5690 3554 3503 3521

0 0 0 1723.8 1738.1 1745.4 1614.9 2487.0 1358.9

Path 3 5596 2288 -525 5693 3505

0 1747.1 2582.0

Figure 2. 7. Computed flow paths in the FEFTRA and NAMMU models. All three paths with the repository and fracture zone R21 (the bedrock model) are presented on top-left. The flow path 1 and 3 are shown on top-right, whereas the flow path 2 is presented at bottom.

17

3 INTEGRATED EC-DFN MODEL IN CONNECTFLOW

Hartley et al. (2001) performed flow simulations for the Olkiluoto site by applying an in­tegrated site and canister-scale model. The simulations were carried out by the finite ele­ment program CONNECTFLOW (Hartley 1996; Holton & Milicky 1997), which couples the NAMMU equivalent continuum (EC) program with the NAPSAC discrete fracture network (DFN) program. In this chapter, the principles of the CONNECTFLOW model are reviewed on the basis of the available material presented in Hartley ( 1996), Bolton & Milicky (1997) and Hartley et al. (2001).

3.1 General

The CONNECTFLOW code consists of both the EC and the DFN models to simulate (constant density) groundwater flow. The two approaches can be applied in distinct sub­regions of the same flow model. Within the DFN submodel the hydraulic properties are defined in terms of stochastic parameters associated with fracture sets, while the deter­ministic parameters are related to the EC medium. The sub-regions are coupled at the interface between the regions by equations expressing the continuity of pressure and the conservation of mass.

The coupling is implemented with additional equations to link nodes on the interface be­tween the sub-regions. In the DFN sub-region fractures are terminated against the surface of the sub-region. Nodes are placed on all fracture intersections and on the trace of frac­tures that intersect the surface of the sub-region. Where these surfaces are at an interface with the EC model, equations link the values at these nodes with nodes in the adjoining porous medium finite-elements. The equations ensure that the pressure on the fracture trace is an interpolate of the pressure in the adjoining element. Likewise, the mass flux through the fracture trace is injected/abstracted from the adjoining element to conserve mass. Thus, pressure in the EC and the DFN media can be computed simultaneously.

Two numeric formulations to represent the exchange of mass at the interface has been included into CONNECTFLOW. In the first (mass-lumping) all the exchange of mass between the sub-regions takes place only at discrete points on the interface. The fracture intersections with the interface are linked to the adjoining continuum finite-elements only at the nodes on fracture intersections. Two nodes are used per fracture intersection, one at either end of the intersection. At each fracture node the pressure is constrained to be an interpolate of the pressure at the coincident point in the EC medium finite element, and the mass passing through the intersection is lumped at the node to act as a point source/sink in the adjoining EC finite element. This method is appropriate for the case where fractures are of similar or smaller length than the EC finite elements at the EC-DFN interface. The second formulation (distributed flux) distributes the exchange of mass along the trace of the fractures on the interface. For long fractures whose intersection crosses several EC finite elements it is more accurate to distribute the mass exchange along the trace of the fracture intersection.

18

For the nested model of Olkiluoto (Hartley et al. 2001) the mass-lumping formulation gave a much better rate of convergence using a preconditioned conjugate gradient method than the distributed mass formulation. This was considered to be due to the mass-lumping approach giving slightly simpler coupling equations, and hence an easier system matrix for the iterative solver. Because the mass-lumping formulation was adequate for the sit­uation and its preferable convergence characteristics it was used for the Olkiluoto model. Some details of the model on the interface between the EC and DFN medium is presented below based on the reports by Hartley (1996) and Holton & Milicky (1997).

3.2 DFN model equations

In the fracture network sub-region the flow equation for the node i is given by

j L Po~ VW; vwiPidn1 - j W;FdOn1 = o, (3.1)

n1 J an1

where n 1 is the fracture network sub-region, an 1 is the set of intersections of the fracture network sub-region with the boundary, p0 is the density of water (constant parameter), T is the fracture transmissivity, 11 is the viscosity of ground water, \l1 i is the linear basis function for the node j on the fracture, Pi is the residual pressure at the node j in the fracture network and F is the consistent mass flux out of the fractures for those fracture that intersect the boundary. The summation in j is taken over the nodes on the fracture network.

By introducing a new mass-flux freedom fi at each node on the boundary at which a Dirichlet (specified pressure) type boundary condition has been set, the mass flux F in Equation 3.1 can be expressed as

F = Lfjwjlan1. j

(3.2)

Thus, on the boundary between the region containing fractures and the region containing porous medium there are two freedoms per node j, the pressure Pi and the flux fi.

3.3 EC model equations

In the porous medium sub-region the flow equation for the node i is given by

Po (j L ~ \1'1/J; \l'lj;iP;dl! - j 'lj;;v · ndOQ) = 0, (3.3)

n J an where n 1 is the porous medium sub-region, an 1 is the boundary of the porous medium sub-region, p0 is the constant density of groundwater, k is the permeability, 11 is }he vis­cosity of groundwater, 'l/Ji is the linear basis function at the EC medium node j, Pi is the residual pressure at the EC medium node j and v is the Darcy velocity and n is the unit outward normal on the boundary of EC medium. The summation in j is taken over the EC medium nodes.

--------------------------------~- - - -

19

3.4 Interface conditions- mass lumping approach

In the mass-lumping approach the conservation of mass is achieved by distributing the mass exchange across the interface over a set of discrete point sources, i.e. the set of fracture nodes on the interface. The substitution of Equation 3.2 into Equation 3.3 gives the conservation of mass across the interface

(3.4)

where Xj is the location of the fracture node j on the interface. The continuity of pressure is handled by interpolating the pressures from the EC model at the fracture nodes on the interface

Pi= L ~j(xi)Pj· j

(3.5)

Equations 3.4 and 3.5 comprise the constraints for the discrete interface problem in the mass-lumping approach.

3.5 Interface conditions- distributed flux approach

In the distributed flux approach the conservation of mass is achieved by distributing the mass exchange across the interface along the trace of the fractures on the EC medium elements they abut. The substitution of Equation 3.2 into Equation 3.3 leads to the con­servation of mass across the interface

I L ~'V'I/JN'I/JAdn- I 'I/J;Fd80. = o. n J an

(3.6)

In this case F is a line source of mass into the EC medium element, in which the node i lies distributed along the lines of intersections with the fracture planes. Thus, the second term in Equation 3.6 can be written as

1 '1/;/I!jfJdl, (3.7)

fracture

where the integral is along the trace of the fractu~e on the interface and the summation is over the fractures that abut the current EC medium element. Consistent to the above treatment of conservation mass the continuity of pressure is handled by the weighted residual approach as follows

I W;(P- P)dl = 0, (3.8)

fracture

20

where i ranges over all the nodes on the edge of the fracture, the integral is along the trace of the fracture on the interface, f> is the pressure in the EC medium

(3.9)

and P is the pressure in the fracture network

(3.10)

Thus, Equation 3.8 can be written as

(3.11)

fracture fracture

Equations 3.6, 3.7 and 3.11 comprise the constraints for the interface problem in the distributed flux approach.

21

4 QUADTREE-BASED INTEGRATED EC-DFN MODEL IN FEFTRA

4.1 Review of current solutions

It has been recognised that the equivalent continuum (EC) and the discrete fracture net­work (DFN) approaches bear complementary advantages and drawbacks on different scales. One of these is the issue of initial and boundary conditions: for the larger (re­gional or far-field or site) scale models it is often easier to define plausible boundary conditions, while, however detailed a smaller (site or near-field or canister) scale DFN model might be inside the modelled area, the results are susceptible to the inherent uncer­tainties of the boundary conditions prescribed over the arbitrary bordering planes of the DFN model.

Thus, boundary conditions for a DFN model are often calculated with an EC model. In the simplest case the EC model saves its results in disk files, and an intermediate tool interpolates this information over the boundaries of a DFN model, which, in turn, is solved by a third software. This approach was used, e.g., in the groundwater flow analyses at Olkiluoto (Lofman 1999; Poteri & Laitinen 1999) performed for the latest safety analyses TILA-99 (Vieno & Nordman 1999).

Unfortunately, the one-way data transfer is particularly unsatisfactory if the DFN model contains internal boundary conditions, like withdrawal of water or injection of a tracer. In this case the DFN model should pass back its computed results along its boundaries to the EC model, which uses them as internal boundary conditions. The EC model should be solved again and this process is repeated iteratively as long as results calculated by them get close enough along the EC-DFN interface so that they could be used as boundary conditions for the final solution of the DFN model. As a matter of course, it is assumed that the iteration converges. If not, it is a sign of, eg., a conceptual inconsistency or a numerical error, which must be resolved before the calculations can go on.

If the data transfer takes place via disk files, only one of the models will reside in computer memory at a time. Denoting the number of nodes in the EC and the DFN models with n and m, respectively, the sizes of the (sparse) system matrices to be solved are n2 or m2 .

In case conservation of computer memory is not of major concern, the same software can handle the two models simultaneously. In this somewhat optimised set-up no disk files are involved and the iteration takes place in the memory (called sequential coupling). However independent of each other, both models reside in computer memory, thus the problem size is n2 + m2 • It must be rather cumbersome to set up two numerical models this way, and we know of no implementation of any of these two-way couplings of the EC and the DFN submodels.

A significant step up was implemented in CONNECTFLOW (Hartley 1996; Holton & Milicky 1997; Hartley et al. 2001 ), where the two models exist at the same time in computer memory, and the two-way coupling is achieved with equations linking the EC model to the DFN model at their common interface. Thus- at the expense of a larger system matrix - results for both models are calculated simultaneously. If the number

22

of nodes at the interface is i, the size of the (sparse) system matrix to be solved is ( n + m+ i) 2 . Note, however, that part of the EC submodel covers the area occupied by the DFN submodel as well, and results calculated for these so called "dead elements" are totally useless. Considering that at the EC-DFN interface the density of nodes in both models should be somewhat similar (at least not orders of magnitudes different), unless a sophisticated adaptive mesh generator is used for the EC model, the proportion of these dead elements and nodes may become considerable. In case in the EC submodel the number of nodes beyond the-EC-DFN interface is nout and within it is nin, the problem size, again, is: (nout + nin +m+ i) 2

•

When the modelled phenomenon requires the use of internal boundary conditions inside the DFN submodel, these boundary conditions are prescribed on the nodes of the DFN submodel, not on the "dead" nodes of the EC submodel falling inside the DFN submodel. Thus, results calculated for the EC-DFN interface are influenced by the external boundary conditions of the EC submodel, the internal boundary conditions of the DFN submodel, and the material properties of the "dead" EC elements inside the DFN submodel. Even though it is possible to define the material properties of these "dead" EC elements such that they would not participate in the flow, their sheer presence constitutes a source of modelling and numerical errors.

As a matter of course, the equations that act as links between the EC and the DFN models strongly depend on the phenomenon modelled: different equations are needed for ground­water flow, solute transport and heat transfer. In CONNECTFLOW only the equations linking pressure across the EC-DFN interface are implemented at the moment.

Thus, when the geology of a site requires, and the amount and quality of data allows a study based on DFN modelling, the boundary conditions are either obtained with a rudimentary one-way data interpolation program, or with a rather cumbersome, two-way, iterative process. Alternatively, a fully coupled solution of the EC and DFN submodels is feasible, albeit involving a significant amount of development work and computational resources even for one type of result quantity.

4.2 Objectives and basic concepts

The main objective of this work was to test a concept (first with a pilot project in 2D) that addresses and eliminates the issues outlined above. Further goals were to gain insight into DFN modelling technology, using approaches implemented in FracMan (Dershowitz et al. 1998), and with special emphasis placed on the extensibility of the code to 3D.

The basic idea of this concept was that the finite elements at the EC-DFN interface would be connected already at the mesh level, not at the equation level. The new approach was based on a previously developed adaptive and recursive mesh generation code FEF­

TRA/ quad tree, which subdivides the modelled area into quadrats around the areas of interest (points or sections) and then triangulates the quad tree grid so that the mesh would be appropriate for finite element analysis (see FEFTRA (2001) for details and examples). As the algorithm supports inherently the efficient local refinement of the mesh without

23

high computational costs, it seemed applicable also to the integration of the DFN subre­gion into the EC model. Especially, the FEFTRA/ quad tree code enables a natural and straightforward coupling of the two different models at the mesh level by way of common nodes, virtually eliminating the problems associated to data transfer across the EC-DFN interface. Furthermore, with linking the EC and DFN submodels at the mesh level, the in­tegrated modelling of other physical phenomena such as solute transport and heat transfer would become feasible as well.

The generation of the quad tree-based integrated model consists of three main steps. First, the EC model is discretised with the adaptive mesh generator FEFTRA/ quad tree using local refinement of arbitrary depth around the EC-DFN interface in order to attain similar nodal density along the EC side of the interface as in the DFN submodel. Inside the area characterised with the DFN submodel no EC elements are finally saved. Thus the problem size remains, using the notation introduced above, ( nout + m )2

• Secondly, the DFN submodel is built from fracture sets of stochastic geometry and hydrologic prop­erties with the extended FEFTRA/ quad tree code. Thirdly, the two submodels, both consisting of finite elements and nodes, are connected at their common nodes falling onto their common interface, and the resulting mesh as well as the generated hydrologic prop­erties are saved for our FE analysis code FEFTRA/ sol vi t. Having defined the initial and boundary conditions for the external boundaries of the EC model, the analysis code with its mixed element library can compute hydraulic head or any other result quantity (pressure, concentration, temperature or Darcy velocities) for the whole mesh, including both the EC and DFN submodels, at the same time.

4.3 Necessary input

The complementary input data to the FEFTRA/ quad tree code relate to both the EC and the DFN submodels. It is necessary to specify the fundamental hydrologic properties (transmissivity, width, storativity and effective porosity) of the matrix and the determin­istic fracture zones of the EC model:

# Vertical quadrilateral nvertices = 4

0. 0. 0. # 1st corner x,y,x 0. 0. -1500. # 2nd corner x,y,z

3000. 0. -1500. # 3rd corner x,y,z 3000. 0. 0. # 4th corner x,y,z 5.0D-9 1.0 # default transmissivity l.OD-9 # default storativity l.OD-4 # default porosity

for matrix,

# Eight sections, extending beyond the model boundaries nsect = 8 name = repository

1000. 0. -500. 2000. 0 . -500. 11

# pl (x, y)

# p2(x, y) # ipid(repository)

default layer width

24

5.0D-8 5.0 # transmissivity & width l.OD-7 # storativity l.OD-4 # porosity

# this line must be empty

Data describing the DFN submodel are organised into global and fracture set specific parameters, most of which default to some sensible value.

The boundaries of the DFN submodel, which will also be the internal boundaries of the EC submodel, are given as the corners of a polygon. The EC element size over these boundaries can be controlled with the recursion level FEFTRA/ quad tree will use when refining the adaptive mesh at this polygon's sides:

# DFN submodel area dfn_border_vertices = 4

945. 0.0 -480. # 1st corner x,y,z 985. 0.0 -480. # 2nd corner x,y,z 985. 0.0 -520. # 3rd corner x,y,z 945. 0.0 -520. # 4th corner x,y,z

dfn_border_level = 12

True DFN global variables relating to the DFN submodel are the number of the generated fracture sets, nodal tolerance, transmissivity cut-off and the initialisation of the random number generator:

# some global variables over the DFN submodel no_of_fracture_sets = 2

closest_nodes = l.OD-6

transmissivity_cutoff = l.OD-16

# initialise the LAPACK/DLARAN random number generator #with four integers in [0 .. 4095], the fourth must be odd

iseedl 12 iseed2 113 iseed3 3124 iseed4 1917

The geometry of the fracture sets consisting of lD sections are defined with their genera­tion region inside the DFN submodel, orientation and size distributions, fracture intensity and cross section area available for the flow. The spatial model (the distribution of the ran­dom seeds from which the fractures are developed) is assumed to be Baecher: fractures occupy their generation region randomly (Dershowitz et al. 1998).

25

The hydrologic properties of the fracture sets are specified with a probability distribution:

BEGIN_SET Local

# geometric properties generation_region_vertices 5 # defaults to dfn_borders polygon

980.0 0.0 -475.0 # 1st corner x,y,z 990.0 0.0 -475.0 # 2nd corner x,y,z 990.0 0.0 -490.0 # 3rd corner x,y,z 980.0 0.0 -525.0 # 4th corner x,y, z 965.0 0.0 -525.0 # 5th corner x,y,z

orientation_distribution Normal # compulsory Ori _Mean -330.0 # compulsory [degrees]

Ori StdDev 4.9 # compulsory [degrees]

size_distribution Normal # defaults to Normal Size_Mean 5.0 # defaults to 20% of generation

Size StdDev 2.0 # defaults to 25% of Mean -

intensity_type NOF # defaults to NOF=200 fractures fracture_intensity 800 # compulsory if intensity_type

width 1.0 # defaults to 1.0 m

# hydrologic properties transmissivity_distribution LoglONrm # defaults to LogNrm

Tr_Mean -11.0 # defaults to -7.0 Tr StdDev 1.0 # defaults to 0.5 -

storativity_distribution LogNrm # defaults to LogNrm St_Mean -7.0 # defaults to -9.0

St StdDev 0.5 # defaults to 1.0 -

porosity_distribution LogNrm # defaults to LogNrm Poro_Mean -4.0 # defaults to -5.0

Poro StdDev 2.0 # defaults to 0.2 -

END_SET

The modular structure of FEFTRA/ quad tree allows the incorporation of further de­tails, like alternative spatial models, different distributions, fracture age specific informa­tion like truncation etc.

4.4 The process of generating the DFN submodel

Prior to going into the details of actually connecting the DFN submodel to the EC sub­model, the steps to be taken that far are reviewed.

First, an adaptive quadtree mesh is generated, taking the boundaries of the DFN submodel as line-like seeds of refinement. For reasons accounted for below, it is often necessary

region

given

26

I 7 r-

I - lL I 1 L I ...... ~'---. - I 1\ I / 1\

-r----- " I J \ I -+- ........ 'I V \

~~ --r- f...J.. 1/ \ 11 --.., r- .....

1/ / - r- !'-- ......, I I 1-1--I / r- I

- -t ~ -I V

- ./... I 1/ u. -

Figure 4.1. Subdivision of the modelled area into quadrats for the demonstration case (Chapter 5 ). The mesh around the fracture zones, repository and, especially DFN region has been refined.

to develop smaller EC elements at these boundaries than for other line-like seeds (e.g., deterministic fracture zones), thus a higher recursion level can be given (Figure 4.1). Secondly, the quadtree mesh is triangulated, near-boundary nodes are moved onto the lines of boundaries (Figure 4.2), and the shape of deformed elements is fixed with the Lagrange correction. Thirdly, the DFN submodel is developed within its boundaries. The DFN submodel is built from fracture sets, of which parameters are estimated or identi­fied from core logs, trace maps, BHTV images etc. The genesis of a fracture set is the following:

• The generation region of the set is defined,

• generation points called seeds are laid down in the generation region according to the set's spatial model (only Baecher is implemented),

• orientation and

• size data are computed from the given distributions, from which

• the coordinates of fracture endpoints are determined,

• the fracture is truncated at the borders of the generation region,

• stochastic hydrologic properties are assigned to the fracture.

27

Figure 4.2. Triangulated quadtree mesh for the demonstration case (Chapter 5 ).

As with any DFN software in general, the user should be aware of the fact that the dimen­sions of the generation region and the size distribution of the fractures are not completely independent of each other. If a large portion of the fractures is truncated off from their original size by the generation region, the specified size distribution cannot be maintained.

Having generated all the sets, the portions of the deterministic fracture zones crossing the area of the DFN submodel are added to the fractures, with their deterministic hydraulic properties. Subsequently fractures with transmissivities below the transmissivity cut-off are taken off.

At this point the geometry of all fractures inside the DFN submodel area has been devel­oped. Obviously, fractures, that are not connected to the boundaries of the DFN submodel and do not intersect other fractures that do, will not participate in the flow. These fractures (called floaters) are not converted into finite elements. Fractures directly or indirectly connected to the boundaries of the DFN submodel will have to be found, and they will be found by "colouring" or "painting" them along the conversion. In the beginning of this conversion process, fractures directly connected to the boundaries of the DFN submodel are registered and all the fractures are re-ordered such that the registered ones come first - mimicking that a tracer or paint would be penetrating from the boundaries into the fractures (Figure 4.3).

In converting the fractures into finite elements, the intersection points between all regis­tered fractures are searched. When a painted fracture intersects a non-painted one, the resulting fractures will also be coloured and re-ordered such that all painted fractures come first. A fracture is converted into a ID finite element if it is registered (meaning it is directly or indirectly connected to the boundaries of the DFN model and thus, with the EC elements also to the boundary conditions), and no more intersection points can be found with other fractures. The conversion process stops when no more registered fractures are available (Figure 4.4).

28

Figure 4.3. Initial state of fractures at the beginning of converting them into finite ele­ments. Fractures that are directly connected to the EC-DFN interface (blue) and the ones having no direct connection to the interface (green) are registered.

Figure 4.4. Final state of fractures in the end of converting them into finite elements. Fractures that are directly or indirectly connected to the EC-DFN interface (blue) have been converted into finite elements, while the remaining fractures without connection to the interface (green) have not.

29

4.5 Connecting the DFN submodel to the EC model

When the lD finite elements making up the DFN submodel are all developed, their end nodes are checked for being fallen onto one of the polygon sections of the DFN submodel boundary. In case a DFN node is detected to be on the boundary (Figure 4.5), the lD DFN element is reconfigured such that its boundary node is replaced with the closest EC node on that boundary section (at the corner points of the DFN subregion there may occur some DFN elements of which both nodes are re-configured). Finally, EC elements falling inside the DFN-submodel as well as outside the modelled area are discarded, the remaining nodes and elements are renumbered and the mesh as well as the property lookup table are saved for the analysis code FEFTRA/ sol vi t. Thus, generating the DFN submodel, putting it into the pre-defined niche inside the EC submodel and connecting the two are integral parts of the unattended automatic mesh generation process, with all implied advantages.

Note, that in case nodes are too scarce along the EC-DFN interface, more DFN finite ele­ments may be connected to a single EC node than would be desired. Also, the orientation of these fractures may significantly deviate from their original, thus corrupting the given orientation distribution. This can be prevented by prescribing small enough EC elements along the interface with high enough recursion level in FEFTRA/ quad tree.

Figure 4.5. The generation of the DFN submodel and the coupling of the EC and DFN models. The fractures directly or indirectly connected to the EC-DFN interface (blue bars on the left) are converted into finite elements (red bars on the right), while the fractures having no connection to the interface (green bar on the left) are not. If the fracture is directly connected to the interface (left) i.e. to the EC finite elements (black triangles), the boundary node of the DFN element (left end of black dashed line) is replaced with the closest EC node on the interface (right end of black dashed line).

30

4.6 Computational aspects, implementation details & QA

As mentioned earlier, the DFN extension to FEFTRA/ quad tree was developed with the potential extension of the 3D FEFTRA/ oc tree code in mind. This primarily meant the design of most of the algorithms and data structures so that they could be used in three dimensions only with the obvious modifications.

Even though being a prototype research tool in a pilot project, FEFTRA/ quad tree was designed to be efficient, building upon the success of the concepts implemented in its original code base. With no regard to conserving computer memory, a large internal database is built and maintained for the mesh and the fracture sets, which allows fast integer operations in data lookup and requires only a minimal amount of floating-point calculations. To further enhance the efficiency of the latter, the arbitrarily positioned plane of the 2D model is transformed into the x - y plane and all subsequent operations are performed there with only two coordinates of the geometrical objects. Prior to saving the final mesh the whole model is moved back to its original position using the inverse geometrical transformations.

Note that while the step of connecting the DFN submodel to the EC-DFN interface is eventually simple and very fast, it is strongly supported by and is only possible with the previously set up internal databases in FEFTRA/ quad tree.

The code performs numerous independent checks upon its partial results in all phases of the mesh generation process. The outcome of these diagnoses is either an on-screen message or the actual state of the mesh or of some other geometrical object saved for the visualisers GLE (Pugmire & Mundt 1995) or GMV (Ortega 2001), rendering debugging and QA of the code easier. Some of these checks can be switched off when the user is confident about the correctness of the mesh and performance is of higher priority.

FEFTRA/ quad tree was developed using modem software development concepts and tools (make system, version control and test cases). The robustness of the code was tested with four FORTRAN77 and two FORTRAN90 compilers on two different com­puter architectures (Linux and Silicon Graphics), as well as the static source code checker FTNCHEK (FTNCHEK 2001). As its name shows, whenever possible the FEFTRA/ quad tree pre-processor makes extensive use of functions from the FEFfRA package.

31

5 EXAMPLE OF THE QUADTREE-BASED INTEGRATED EC-DFN MODEL

Besides using several test cases in the development phase for diagnostic purposes, this study included the elaboration of a demonstration case as well. The involved simulations were to showcase the ability of the developed quad tree-based integrated model and the FEFfRA finite element solver to handle the realistic cases that could be encountered in foreseeable modelling tasks. The case consisted of

• the construction of a finite element mesh with integrated EC and DFN regions,

• the generation of the transmissivity distribution, and

• the computation of groundwater flow (hydraulic head and the Darcy velocity).

As Olkiluoto has been chosen to be the primary site for the repository in Finland and the latest site-specific groundwater flow analyses for Olkiluoto were carried out by app­lying the two nested and separate models on different scales (the EC model on the site scale (Lofman 1999) and the DFN model on the canister scale (Poteri & Laitinen 1999)), the demonstration case was chosen to be based on the Olkiluoto site. Thus, when building the hypothetical case, the following input were set as close to the ones in the previous site and canister scale analyses for Olkiluoto as possible:

• the dimensions of the modelled area,

• the deterministic fracture zones,

• the number of fracture sets to build the DFN submodel,

• the orientation and size data to create fractures,

• fracture intensity,

• the range of values describing the hydrologic properties, and

• the characteristics of boundary conditions.

The case concerns a steady state groundwater flow (with fresh water density) deep in the bedrock below an island surrounded by sea (Figure 5.1). The bedrock is assumed to contain seven deterministic planar fracture zones and the horizontal repository. The modelled 2D domain is a vertical cross-section with a width of 3.0 km and a depth of 1.5 km. The island is located at the centre of the modelled domain. The groundwater table (maximum 10 m) follows the parabolic curve

h(x) = 10(1- (x ~;~00 )2), 750::::; x::::; 2250. (5.1)

The specified hydraulic head is employed as boundary condition on the top of the mod­elled domain, whereas other boundaries are assumed to be impermeable to flow. As the purpose of the canister scale modelling (Poteri & Laitinen 1999) has been to characterise the groundwater flow conditions and transport properties in the vicinity of the disposal

32

lSOOm

Figure 5.1. Schematic description of the demonstration case for the integrated EC-DFN model in 2D.

canister and the nearest fracture zones, a square shaped DFN subarea ( 40 m x 40 m) was placed between the zones R1 and R2 near the repository. In addition, the subhorizontal zone R7 is assumed intersect the DFN region.

The finite element mesh with the integrated EC and DFN elements (Figure 5.2) was gen­erated adaptively with FEFTRA/ quad tree program (input file for the demonstration case is presented in Appendices A-C). The largest element size in the EC region was li­mited to 100 m. The local refinement of the mesh in the EC region resulted in the element size of about 10 m near the seven deterministic fracture zones and the horizontal repos­itory (recursion level 8 in FEFTRA/ quad tree), whereas the element size was about 1 m around the DFN submodel (recursion level 12 in FEFTRA/ quad tree). The EC region consisted of 18924 triangular and bar elements for the matrix and fracture zones. The transmissivities of the deterministic fracture zones and the matrix were based on the site scale model of Olkiluoto (Lofman 1999) and varied exponentially with depth on two ranges (T(zones) = 10-7 .. 10-3 m2 /sand T(matrix) = 10-12 .. 10-7 m2 js).

The DFN submodel was created from two fracture sets (Table 5.1), which were based on the DFN models by Poteri & Laitinen ( 1999) for the Olkiluoto site. Initially the two sets included a total of 1600 individual fractures. However, as all the fractures were not connected directly or indirectly to the boundaries of the DFN submodel, only 1509 fractures were split at their intersection points and converted into 12701 bar elements. All the created finite elements inherited the stochastic hydrologic properties that were assigned to their "parent" fractures.

The geometric and hydrologic properties of the fractures were based on the canister scale model of Olkiluoto (Poteri & Laitinen 1999). The orientation and size distributions were normally distributed, whereas the transmissivity followed 1 0-based lognormal distribu­tion. The standard deviation of 1.0 was selected for the simulations, because higher values

----------------------------- -----

33

Parameters Set#l Set#2 orientation distribution Normal Normal mean [degrees] -330.0 -300.0 stddev [degrees] 4.9 6.2 size distribution Normal Normal mean 5.0 5.0 stddev 2.0 2.0 No of fractures 800 800 transmissivity distribution log10normal log10normal mean [in log10 space] -11.0 -11.0 stddev [in log10 space] 1.0 1.0 porosity distribution lognormal lognormal mean [in log space] -4.0 -5.0 stddev [in log space] 2.0 0.2

Table 5.1. The geometric and hydrologic properties of the stochasticfractures in the DFN submodel.

resulted in numerical problems in the convergence of the iterative solver (conjugate gra­dient solver with Jacobi preconditioning), when solving the hydraulic head and Darcy velocity. The two fracture sets differed from each other only in their the orientation. The transmissivity distribution over the entire model and in a close-up at the DFN subregion is shown in Figure 5.3.

The result quantities of the demonstration case were steady state hydraulic head and Darcy velocity, which were computed with one DFN realisation. The computed hydraulic head (Figure 5.4) shows that the ground water table and deterministic fracture zones are the con­trolling factors in the EC model, while in the DFN submodel the head field follows head distribution at the EC-DFN interface. On the other hand, the Darcy velocity (Figure 5.5) is dependent on the combined effect of hydraulic gradient and transmissivity distribution. The higher transmissivity of the fracture zones and the depth dependency can clearly be seen in the velocity in the EC region. On the other hand, the effect of empty space be­tween fractures and the stochastically distributed transmissivity field can be observed in the DFN subregion. The Darcy velocity vectors in the vicinity of the EC-DFN interface (Figure 5.6) indicate that groundwater in the EC region tend to flow around the DFN re­gion, while most of water flowing through the DFN model flows along the fracture zone R7. On the other hand, in the DFN model, especially above the zone R7, some routes with a higher velocity can be identified.

The demonstration case was also computed assuming the equivalent continuum medium for the DFN region (Appendix D). The mean transmissivity of fractures (Table 5.1) was used for the EC elements at the DFN region. Although the head (Appendix E) and Darcy velocity (appendix F) can not be directly compared to the ones computed with the inte­grated model, they can still be seen to be consistent to each other.

----------------------------------------------- - --- --

34

Figure 5.2. Finite element mesh of the demonstration case for the integrated EC-DFN model in 2D. The mesh consists of 18924 triangular and bar elements for the EC medium (top, matrix and deterministic fracture zones) as well as 12701 bar elements for the DFN subregion (bottom, 1509 stochastically distributed fractures). The green and blue colours in the DFN mesh represent the two fracture sets.

35

lo~0(T) [m2/s]

-3.00D+OO

-6.00D+00

-8.00D+OO

-9.00D+00

-1.00D+01

-1.10D+01

-1.20D+01

-1.30D+01

-1.40D+01

-1.60D+01

Figure 5.3. Transmissivity distribution in the demonstration case for the integrated EC­DFN model in 2D. The transmissivity varies exponentially with depth in the EC region (top) and is lognormally distributed in the DFN subregion (bottom).

36

head [m]

1.00D+01

8.00D+OO

6.00D+OO

4.70D+00

4.40D+00

4.10D+OO

3.80D+OO

3.50D+OO

2.00D+00

O.OOD+OO

Figure 5.4. Computed hydraulic head in the demonstration case for the integrated EC­DFN model in 2D.

37

q [m/s]

1.00D-09

1.00D-10

1.00D-11

1.00D-12

1.00D-13

1.00D-14

1.00D-15

O.OOD+OO

Figure 5.5. Computed Darcy velocity in the demonstration case for the integrated EC­DFN model in 2D.

38

interface betwet:_g. the E~

~/

/

Figure 5.6. Computed Darcy velocity vectors in the demonstration case near the EC­DFN interface. The colour and the contour levels are the same than in Figure 5.5.

--------------------------- - --

39

6 SUMMARY AND DISCUSSION

For the latest safety analyses TILA-99 (Vieno & Nordman 1999) the site-specific ground­water flow analyses at Olkiluoto were carried out by applying the two nested and separate models on different scales. The equivalent continuum (EC) approach was employed on the site scale (Lofman 1999) and the discrete fracture network (DFN) approach on the canister scale (Poteri & Laitinen 1999). The two separate models had a very simple one­way coupling, i.e. the overall flow properties for the canister scale were derived from the results of the site scale modelling. Because the size and modelling approaches between the two models were quite different, the estimation of the boundary conditions was subject to inconsistency and large uncertainties. This work comprised a preliminary study on the integration of site and canister scale models into a single groundwater flow model, which ensures that the flow conditions on the canister scale are consistent with the localised conditions on the site scale.

Hartley et al. (200 1) performed flow simulations for the Olkiluoto site by applying mod­els on different scales. The study consisted of a site scale model testing, in which a steady-state groundwater flow in 3D was computed with NAMMU (NAMMU 2001) and FEFTRA (FEFTRA 2001) program packages for the Olkiluoto site and the results were compared. In this work, the case was documented and examined with regard to the modelling performed with FEFTRA (Chapter 2). The result quantities computed in the test case were pressure along the boreholes KR1-KR5, flow paths starting at three points near the repository, flow rates through a box surrounding the repository and infiltration. The pressures (Figure 2.6), flow rates (Table 2.4 ), infiltration and the final positions and lengths of the flow paths (Table 2.5 and Figure 2. 7) showed an excellent agreement be­tween the FEFTRA and NAMMU models. On the other hand, the travel times of the flow paths computed with FEFTRA were shorter than the corresponding NAMMU times, which can probably be attributed to the different representations of the fracture zones and/or different discretisation in the vicinity of the repository. However, the case verified further the capability of the FEFTRA code to simulate real-life site-scale groundwater flow problems employing 2D elements for the fracture zones in the 3D mesh.

Hartley et al. (200 1) performed flow simulations for the Olkiluoto site by applying an integrated site and canister-scale models. The simulations were carried out by the fi­nite element program CONNECTFLOW (Hartley 1996; Holton & Milicky 1997), which combines the EC and DFN models. In this work, the principles of the CONNECT­FLOW model were reviewed on the basis of the available material presented in Hartley (1996), Holton & Milicky (1997) and Hartley et al. (2001) (Chapter 3). The CONNECT­FLOW model consists of two coupled approaches to model groundwater flow, the DFN and EC approaches, which can be used in distinct sub-regions of the same flow model. The sub-regions are coupled at the interface between the regions by continuity of pres­sure and conservation of mass. The coupling is implemented by adding in extra equations to link nodes on the interface between the sub-regions. The equations ensure that the pressure on the fracture trace is an interpolate of the pressure in the adjoining element. The mass flux through the fracture trace is injected/abstracted from the adjoining element

40

to conserve mass. Due to the coupling the pressure in the EC and DFN media can be computed simultaneously.

The main objective of this work was to develop an integrated EC-DFN model (first as a pilot project in 2D with special emphasis placed on .the extensibility of the code to 3D), in which the two models would already be connected at the mesh level instead of the equation level (Chapter 4). The new approach was based on a previously developed FEFTRA/ quad tree code, which employs an adaptive and recursive algorithm in mesh generation. As the quadtree algorithm supports the efficient local refinement of the mesh without high computational costs, it also proved applicable to the integration of the DFN subregion into the EC model. The FEFTRA/ quad tree code was extended so that it would be able develop a DFN submodel inside the EC model and connect the two sub­models at their nodes falling on the EC-DFN interface. This mesh-level linking of the two submodels is natural and straightforward, and eliminates the problems - seen pre­viously in other solutions- that are associated to the data transfer across the EC-DFN interface. In addition to ground water flow, with linking the EC and DFN submodels at the mesh level, the integrated modelling of other physical phenomena such as solute transport and heat transfer is feasible as well.

The generation of the quad tree-based integrated model consists of three main steps. First, the EC model is discretised adaptively using local refinement around the EC-DFN interface in order to attain similar nodal density along the EC side of the interface as in the DFN submodel (Figure 4.2). Secondly, the DFN submodel is built from fracture sets of stochastic geometry and hydrologic properties (Figure 4.3 and 4.4). Thirdly, the two submodels, both consisting of finite elements and nodes, are connected at their common nodes falling onto their common interface (Figure 4.5), and the resulting mesh as well as the generated hydrologic properties are saved for the finite element analysis. The im­plementation of this concept was relatively straightforward, even if attaining the desired efficiency in generating the DFN submodel cost more effort than expected. The current bottleneck of the mesh generation process, not surprisingly, is the conversion of fractures into 1D finite elements, which is in fact a truly DFN-modelling problem and is indepen­dent of the original task of treating the EC-DFN interface.

The developed quad tree-based integrated model was applied to one demonstration case (Chapter 5), which was based on the latest groundwater flow analyses for the Olki­luoto site. The case concerned a 2D steady state groundwater flow in bedrock containing deterministic fracture zones and a square-shaped DFN model located between the repos­itory and nearest zones (Figure 5.1). The geometry and the hydraulic properties were based on the site scale model by Lofman (1999). The DFN submodel was created from two fracture sets comprising altogether 1509 fractures (Table 5.1). The parameters of the two sets were based on the canister scale model by Poteri & Laitinen (1999). The orientation and size distributions of fracture sets were normally distributed, whereas the transmissivity followed 1 0-based lognormal distribution. The simulation indicated that the integrated model and FEFTRA finite element code could handle the realistic cases that could be encountered in foreseeable modelling tasks. However, when solving the model for hydraulic head and Darcy velocities (with a Jacobi-preconditioned conjugate gradient solver) it was recognised that the contrast in transmissivity values introduced in

41

the model by too high standard deviation (> 1.0 in 1 0-based logarithmic space) values was detrimental to the convergence of the solution.

The extension of FEFTRA/ quad tree with the DFN submodel and connecting it to the EC submodel, as well as slight modifications to the FEFTRA/ sol vi t FE analysis code took two man-months of work. It is suggested that on the basis of these experiences a 3D version of this extension would also be feasible with a similar order of magnitude- some months - work. Most difficulties in the 3D version are anticipated about the generation of the 3D DFN submodel, especially with the representation of fractures and their con­version into finite elements. Less difficult, albeit also new issues are the treatment of 3D directional data, some currently unimplemented features like truncation, commonly used fracture intensity measures or an alternative spatial model, and probably also some tasks about visualising and postprocessing the stochastic 3D results in multiple realisations. Because of the convergence issues pointed out above it may be necessary to upgrade the iterative solver as well.

The solution to the data transfer across the EC-DFN interface presented herein is simple, natural and builds on the functionality of existing tools. Thus, it can be concluded that connecting the two submodels on the mesh level is a viable and efficient alternative to current solutions.

43

REFERENCES

Atkinson, K. E., 1988. An Introduction to Numerical Analysis. John Wiley & Sons, Inc., New York. Second Edition.

Bear, J., 1979. Hydraulics of Groundwater. McGraw-Hill, Israel.

Dershowitz, W., Lee, G., Geier, J., Foxford, T., LaPointe, P., & Thomas, A., 1998. FracMan User Documentation v2.6. Golder Associates, Seattle, Washington.

de Marsily, G., 1986. Quantitative Hydrogeology- Groundwater Hydrology for En­gineers. Academic Press INC, Orlando.

FEFTRA, 2001. FEFTRA TM program package. VTT Energy. http://www.vtt.fi/ene/ye/feftra.

FTNCHEK, 2001. ftnchek- A static analyzer for Fortran 77. http://www.dsm.fordham.edu/ ftnchek.

Hartley, L. J., 1996. CONNECTFLOW (Release 1.0) - User Guide. AEA Technol­ogy, Harwell.

Hartley, L., Hoch, A., & Holton, D., 2001. An integrated approach to ground water flow modelling on different scales. Working Report 2002-33, Posiva Oy, Helsinki.

Holton, D. & Milicky, M., 1997. Simulating the LPT2 and Tunnel Drawdown Ex­periment at Aspo using a Coupled Continuum-Fracture Network Approach. Interna­tional Cooperation Report 97-05, Swedish Nuclear Fuel and Waste Management Co (SKB), Stockholm.

Huyakom, P. S. & Pinder, G. F., 1983. Computational Methods in Subsurface Flow. Academic Press INC, Orlando.

Laitinen, M., 1994. Developing an Iterative Solver for the FEFLOW Package. Tech­nical Report POHJA-2/94, VTT Energy, Espoo. (in Finnish).

Lofman, J., 1999. Site Scale Groundwater Flow in Olkiluoto. Posiva Report POSIVA-99-03, Posiva Oy, Helsinki.

NAMMU, 2001. NAMMU software package, version 7.0.15. AEA Technology­Environment. http://www.aeat-env.com/groundwater/narnmu.htrn.

Ortega, F., 2001. General Mesh Viewer (GMV). Los Alamos National Laboratory, US DOE, http://www-xdiv.lanl.gov/XCM/gmv/GMVHome.html.

Poteri, A. & Laitinen, M., 1999. Site-to-canister Scale Flow and Transport in Hast­holmen, Kivetty, Olkiluoto and Romuvaara. Posiva Report POSIVA-99-15, Posiva Oy, Helsinki.

Pugmire, C. & Mundt, S. M., 1995. GLE User Manual, v3.3h. University of New Zealand, Departement of Scientific and Industrial Research, Information Technology Group, Physical Sciences, ftp://sunsite.unc.edu/pub/Linux/science/visualization/ gle-3.3h-src.tar.gz.

44

Saksa, P., Ahokas, H., Nummela, J. & Lindh, J. 1998. Bedrock Models of Kivetty, Olkiluoto and Romuvaara Sites, Revisions of the Structural Models during 1997. Work Report PATU-98-12, Posiva Oy, Helsinki. (In Finnish).

Vieno, T. & Nordman, H. 1999. Safety Assessment of Spent Fuel Disposal in Hast­holmen, Kivetty, Olkiluoto and Romuvaara- TILA-99. Posiva Report POSIVA-99-07, Posiva Oy, Helsinki.

Aikas, K., (editor), Hagros, A., Johansson, E., Malmlund, H., Sievanen, U., Tolppa­nen, P., Ahokas, H., Heikkinen, E., Jaaskelainen, P., Ruotsalainen, P. & Saksa, P. 2000. Engineering rock mass classification of the Olkiluoto investigation site. Posiva Report POSIVA 2000-08, Posiva Oy, Helsinki.

45 APPENDIX A

###### QUADTREE INPUT FILE FOR THE DEMONSTRATION CASE ###### # # EC model # # flag showing if we'll generate the mesh dynamically

dynamic = false

# maximum size allowed for a quadrat allowed_largest_element = 150.

# Vertical quadrilateral nvertices = 4

0.0 0.0 0.0 0.0 0.0 -1500.0

3000.0 0.0 -1500.0 3000.0 0.0 0.0 5.0D-9 1.0D-9

1.0 # default transmissivity for matrix, default layer width # default storativity

1.0D-4 # default porosity

# Eight fracture zones, extending beyond the model boundaries nsect = 8 name = repository 1000.0DO 0.0 -500.0DO # p1(x,y,z) 2000.0DO 0.0 -500.0DO # p2(x,y,z) 11 # ipid(S1)

5.0D-8 5.0 # transmissivity & width 1.0D-7 # storativity 1.0D-4 # porosity

# this line must be empty name = zone1 900.0DO 0.0 O.ODO # p1(x,y,z) 600.0DO 0.0 -1500.0DO # p2 (x,y, z) 12 # ipid(S1)

4.4D-4 10.0 # transmissivity & width 1.0D-7 # storativity 1.0D-4 # porosity

# this line must be empty name = zone2 1100.0DO 0.0 O.ODO # p1(x,y,z) 1200.0DO 0.0 -1500.0DO # p2 (x,y, z) 13 # ipid(S1)

1.0D-5 10.0 # transmissivity & width 1.0D-7 # storativity 1.0D-4 # porosity

# this line must be empty name = zone3 2000.0DO 0.0 O.ODO # p1 (x,y, z) 1400.0DO 0.0 -1500.0DO # p2 (x,y, z) 14 # ipid(S1)

1.0D-5 10.0 # transmissivity & width 1.0D-7 # storativity 1.0D-4 # porosity

# this line must be empty

(CONTINUES ... )

(CONTINUED ... )

name = zone4 2400.0DO 0.0 2600.0DO 0.0 15

4.4D-4 10.0 1.0D-7 1.0D-4

name = zoneS 2600.0DO 0.0 1600.0DO 0.0 16

4.4D-4 10.0 1.0D-7 1.0D-4

name = zone6 O.ODO 0.0 3000.0DO 0.0 17

4.4D-4 10.0 1.0D-7 1.0D-4

name = zone7 600.0DO 0.0 1400.0DO 0.0 18

4.4D-4 10.0 1.0D-7 1.0D-4

# # DFN model #

O.ODO -1500.0DO

O.ODO -1500.0DO

-SOO.ODO -1100.0DO

-350.0DO -700.0DO

47 APPENDIX 8

# p1 (x,y, z) # p2 (x,y, z) # ipid(S1) # transmissivity & width # storativity # porosity # this line must be empty

# p1(x,y,z) # p2 (x,y, z) # ipid(S1) # transmissivity & width # storativity # porosity # this line must be empty

# p1 (x,y, z) # p2 (x,y, z) # ipid(S1) # transmissivity & width # storativity # porosity # this line must be empty

# p1(x,y,z) # p2(x,y,z) # ipid(S1) # transmissivity & width # storativity # porosity # this line must be empty

# initialise the random number generator for LAPACK's DLARAN #with four integers in [0 . . 4095], the fourth must be odd

#

iseed1 12 iseed2 iseed3 iseed4

113 3124 1917

DFN submodel area dfn_border_vertices 945. 0.0 -480. 985. 0.0 -480. 985. 0.0 -520. 945. 0.0 -520.

4

dfn_border_level = 12 no_of_fracture_sets = 2 closest_nodes = 1.0D-6 transmissivity_cutoff

(CONTINUES ... ) 1.0D-16

49 APPENDIX C

(CONTINUED ... )

# The first set overrides all default values and thus the user defines # all geometric as well as hydrologic properties

BEGIN_SET Global # geometric properties

orientation_distribution Ori_Mean

Ori_StdDev

size_distribution Size_Mean

Size_StdDev

mode truncation

intensity_type fracture_intensity

width termination

# hydrologic properties transmissivity_distribution

Tr_Mean Tr_StdDev

storativity_distribution St_Mean

St_StdDev

END_SET

porosity_distribution Poro_Mean

Poro_StdDev

BEGIN_SET Local # geometric properties

orientation_distribution Ori_Mean

Ori_StdDev

size_distribution Size_Mean

Size_StdDev

intensity_type fracture_intensity

# hydrologic properties transmissivity_distribution

Tr_Mean Tr_StdDev

END_SET

Normal -330.0

4.9

Normal 5.0 2.0

random true

NOF 800

1.0 20.0

# compulsory # compulsory [degrees] # compulsory [degrees]

# defaults to Normal # defaults to 20% of DFN region # defaults to 25% of Mean

# defaults to centre # defaults to false

# defaults to NOF=200 fractures # compulsory if intensity_type given

# defaults to 1.0 m # defaults to 0.0 %

Log10Nrm # defaults to LogNrm -11.0 #defaults to -7.0

1.0 # defaults to 0.5

LogNrm -7.0

0 . 5

LogNrm -4.0 2.0

Normal -300.0

6.2

Normal 5.0 2.0

NOF 800

# defaults to LogNrm # defaults to -9.0 # defaults to 1.0

# defaults to LogNrm # defaults to -5.0 # defaults to 0 . 2

# compulsory # compulsory [degrees] # compulsory [degrees]

# defaults to LogNrm # defaults to 20% of generation region # defaults to 25% of Mean

# defaults to NOF=200 fractures # compulsory if intensity_type given

Log10Nrm # defaults to LogNrm -11.0 #defaults to - 7.0

1.0 # defaults to 0.5

51 APPENDIX D

Finite element mesh of the demonstration case (EC model)

53

Hydraulic head in the demonstration case (EC model )

APPENDIX E

head [m]

1.00D+01

8.00D+OO

6 .00D+OO

4.70D+OO

4.40D+00

4.10D+00

3.80D+00

3.50D+OO

2.00D+OO

O.OOD+OO

55

Darcy velocity in the demonstration case (EC model)

APPENDIX F

q [m/s]

1.00D-09

1.00D-10

1.00D-11

1.00D-12

1.00D-13

1.00D-14

1.00D-15

O.OOD+OO